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Abstract

One of the most challenging curves to render in the class of conic sections is the ellipse that is rotated at some arbitrary angle. This disclosure represents a summary of material presented in [1].

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United States

Language

English (United States)

This text was extracted from a PDF file.

This is the abbreviated version, containing approximately
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Efficient Rotation and Rendering of Ellipses

One of the most challenging curves to render in the class
of conic sections is the ellipse that is rotated at some arbitrary
angle. This disclosure represents a summary of material presented in [1].

It is possible to mathematically represent conic sections as rational,
quadratic, Bezier curves. The technique of using rational curves was adapted
from [2]. This referenced article contains the parametric equations for a circle. 1
contains a derivation of these parametric equations. Based on that derivation, a
derivation for an orthogonally oriented ellipse is also shown in [1]. Orthogonally
oriented means that the major and minor axes of the ellipse are orthogonal with
respect to the X and Y axes of Cartesian Space.

Based on this derivation, the control points are derived in a manner similar to [2]. The scaled forward difference equations to be used to render the curve are
derived in [1]. Forward difference methods are not new, but they used floating
point arithmetic. Scaling the forward difference equations into the integer domain
means that the error term can be precisely controlled and integer arithmetic is
inherently faster than floating point. Not counting the setup cost or the cost of
looping, the next X and Y coordinate pair is calculated with ten integer additions,
two single bit right shifts, and two integer divisions.

That the control point definitions will yield a mathematically correct ellipse is
also shown in [1]. The Scaled Integer Forward Difference Method approximates
the curve of the ellipse by computing discrete points along the curve and
connecting them with straight line segments. A couple of tricks are used to make
the rendering very efficient.

The equations for the control points are based on an orthogonal ellipse, but
the goal is to render an ellipse that is...